On weighted multilinear polynomial averages in finite fields

TL;DR

This paper investigates weighted multilinear polynomial averages over finite fields, establishing norm control and deriving asymptotic formulas for multidimensional rational function progressions in subsets of finite field vector spaces.

Contribution

It introduces new $u^s$-norm control techniques for weighted multilinear averages and applies them to count rational function progressions in finite fields.

Findings

Established $u^s$-norm control for weighted multilinear averages.

Derived asymptotic formulas for rational function progressions.

Extended understanding of polynomial averages in finite field combinatorics.

Abstract

We study the weighted multilinear polynomial averages in finite fields. The essential ingredient is the $u^s$-norm control of the corresponding weighted multilinear polynomial averages in finite fields, which is motivated by Ter\"av\"ainen \cite{T24}. As an application, we prove an asymptotic formula for the number of the following multidimensional rational function progressions in the subsets of $\mathbb{F}_p^D$: \[ \textbf{x}, \textbf{x}+ P_1(\varphi(y))v_1,\cdots, \textbf{x}+ P_k(\varphi(y))v_k, \] where $\mathbb{V}=\{v_1, \cdots, v_{k} \in \mathbb{Z}^D\}$ is a collection of nonzero vectors, $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{Z}[y]\}$ is a collection of linearly independent polynomials with zero constant terms, and $\varphi(y) \in \mathbb{Q}(y)$ is a nonzero rational function.